Roughening in dimer models
Random matrix statistics and surface fluctuations
Tid: Fr 2023-01-27 kl 13.00
Respondent: Scott Mason , Matematik (Avd.)
Opponent: Professor Alexey Bufetov, Leipzig University, Germany
Handledare: Professor Kurt Johansson, Matematik (Avd.)
The field of mathematical statistical mechanics sits at the intersection of probability theory and mathematical physics. It consists of the rigorous analysis of models in statistical mechanics, such as dimer and lattice models - of which the Ising model is a classical example. The connections between this field and other areas of mathematics is enormous and growing, indeed, the study of such models and their properties of interest has been connected to objects from special function theory, random matrix theory, combinatorics, statistical/quantum field theory, etc. and it is in this field that this dissertation lies. In particular, this dissertation focuses on phase transitions of dimer models, where the corresponding surface fluctuations vary between rough and smooth phases. We investigate these transitions in a variety of novel contexts: we find a description of how correlations between dimers change across a spatial rough-smooth transition, and we show that a microscopically defined path at this boundary is described by a stochastic process appearing in random matrix theory. We also investigate the height field fluctuations of a near-critical dimer model, which we show is described by a non-conformal integrable quantum field theory. This thesis, which is a compilation thesis, consists of an introduction and three original research papers. In particular;
In the first section of the introduction we cover background material related to determinantal point processes, that is, point/particle processes whose correlation functions are given by determinants. We do this by reviewing basic theory and giving a number of relevant examples. In the second section, we cover background material on dimer models. We see that dimer models form a determinantal point process, and we cover basic theorems regarding their integrability and construction of some infinite Gibbs measures. In the final, third section, we give background material on the models which appear in papers A-B.
In paper A, we study correlations between dimers across a spatial transition region of a large, but finite two-periodic dimer model on the Aztec diamond graph. We show that if the dimers are situated in the rough phase but are very close to the smooth phase of the model, their correlations initially decay exponentially. As the distance between the dimers increases this decay can transition to a constant decay, and then transition to a power law decay once far enough apart. The results also apply to an infinite dimer model with weights induced to give the same average slope of the associated height function.
In paper B, we study the limit of the two point correlation function of a height field when the associated parameter of the model (a chemical potential) is close to a rough phase, called a near-critical dimer model. We show that the two point correlation functions limit to those of the sine-Gordon field from integrable quantum field theory. We do this via direct asymptotic analysis of the double contour integral formula which defines the Gibbs measure of the dimer model. This paper also contains a short mathematical review of a sense in which the sine-Gordon field exists.
In the final paper, paper C, we investigate a collection of paths, with a microscopic definition, on the two-periodic Aztec diamond dimer model. We prove there is a last path at the rough-smooth boundary which converges to the Airy process, when the width of the rough region becomes small in the mesh limit. This proves a previous conjecture in a simpler setting. As a byproduct of the analysis, we see that dimers are described by an extended discrete Bessel kernel when the rough phase is only of microscopic width.